Show that Φ(ϕ)=e^i mr ϕ=Φ(ϕ+2 π) (that is, show that Φ(ϕ) is periodic with period 2πif and only

step 1 So we're told this is a function of phi here. Capital Phi is a function of little phi, which is

Show that Φ(ϕ)=e^i mr ϕ=Φ(ϕ+2 π) (that is, show that Φ(ϕ) is...

Show that $\Phi(\phi)=e^{i m_{r} \phi}=\Phi(\phi+2 \pi)$ (that is, show that $\Phi(\phi)$ is periodic with period 2$\pi$ if and only if $m_{l}$ is restricted to the values $0, \pm 1, \pm 2, \ldots$ (Hint: Euler's formula states that $e^{i \phi}=\cos \phi+i \sin \phi . )$

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Key Concepts

Quantization of Angular Momentum

The requirement that the wavefunction be periodic with a period of 2? forces the exponent in e^(i m ?) to satisfy the condition e^(i m (? + 2?)) = e^(i m ?), which in turn restricts m to integer values. This discrete set of allowed values for m is a manifestation of angular momentum quantization in quantum mechanics.

Euler's Formula

Euler's formula, which expresses e^(i?) as cos(?) + i sin(?), serves as a bridge between complex exponentials and trigonometric functions. This connection is essential when analyzing functions that describe periodic behavior, allowing one to interpret the effects of phase shifts and rotations in a clear, mathematical manner.

Periodic Boundary Conditions

Periodic boundary conditions require that a function repeats its values exactly after a certain interval, ensuring consistency and single-valuedness in physical contexts. In problems involving angular variables, this implies that the wavefunction must have the same value when the angle ? is increased by 2?, constraining the structure of the solution.

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